# Questions tagged [cauchy-schwarz-inequality]

The Cauchy-Schwarz inequality states $|\langle x,y \rangle |\leq ||x||\cdot ||y||.$ Use this tag for questions related to the CS inequality and its applications.

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Let $\chi(s)=\int_{0}^{1}x(t)^{s}f(t)dt$, where $x(t)$ and $f(t)$ are real valued continuous functions for $t\in[0,1]$, and $f(t)\geq0$. Is it possible to show that $\left(\chi(0)\chi(2)-\chi(1)^{2}... 1answer 164 views ### Is it possible to find$f$such that :$f$is absolutely integrable,$f'$is absolutely integrable and such that$f$is not$1/2$-Hölder I am trying to find a function$f: \mathbb{R}^+ \to \mathbb{R}^+$that fullfils the following conditions $$f \in \mathcal{C}^1(\mathbb{R}^+,\mathbb{R}^+)$$ $$\int_{\mathbb{R}^+} f \in \... 1answer 223 views ### On the Cauchy-Schwarz Inequality for trace function of random matrices In the deterministic case, for two matrices A and B with appropriate matrices, we know that$$tr((A^{T}B)^{2})\leq tr(A^{T}A)tr(B^{T}B)$$which is the trace form of Cauchy-Schwarz-Inequality (CSI).... 3answers 473 views ### A probabilistic angle inequality Conjecture: There is a universal constant c such that for any fixed nonzero real vector q of any dimension n and any random vector p of the same dimension n with independent components ... 1answer 334 views ### An inequality involving a sum of power terms I am currently working in a problem in Information Theory and I came across a difficult inequality. After many attemps, I simplified the inequality, which now looks at follows. Consider a positive ... 2answers 359 views ### An alternative proof of Bayesian Cramer-Rao My question is: Are there an alternative proof of Cramer-Rao lower bound that does not use Cauchy-Swartz inequality? Let me outline the classical proof and explain why I am interested in this ... 1answer 122 views ### Upper bound of \frac{\sum_i c_ia_ie_i}{\sum_i d_ib_if_i}? Let \sum_i c_i =\sum_i d_i=1, where c_i,d_i \ge 0. Assume that \frac{\sum_i c_ia_i}{\sum_i d_ib_i} \le \epsilon_1 and \frac{\sum_i c_ie_i}{\sum_i d_if_i} \le \epsilon_2, where a_i,b_i,e_i,f_i ... 0answers 221 views ### Does the Cauchy–Schwarz inequality imply 2-positivity? Recall the following generalisation of Cauchy–Schwarz. Theorem. Let f\colon \mathscr{A} \to \mathscr{B} be a linear 2-positive map between C^*-algebras. Then for all a,b \in \mathscr{A} we ... 1answer 127 views ### Majorization of cyclic products Let k,m,n\in\mathbb N such that n>k. For a partition \alpha=(\alpha_1,\dots,\alpha_k)\vdash m with \alpha_1\ge\dots\ge \alpha_k>0 and nonnegative x_1,\dots,x_n define x^\alpha :=\... 2answers 381 views ### Does this simple inequality have a name? Let x_{1},\ldots,x_{n} be nonnegative numbers such that m \leq x_{i} \leq M. Let$$ S=\sum_{i=1}^{n}{x_{i}} $$and$$ Q=\sum_{i=1}^{n}{x_{i}^{2}}. $$Then$$ Q \leq S(M+m)-nMm. $$This has ... 0answers 1k views ### A stronger Cauchy-Schwarz inequality for traces of compression matrices Assume that A and B are contractions, so I-AA^T and I-BB^T are positive-definite matrices. Let C=(I-AB^T)^{-1}(I-AA^{T})(I-BA^{T})^{-1}, and show that:$$Tr\left(\frac{1}{1-AA^T}\right)... 1answer 211 views ### A homogeneous but slightly asymmetric inequality I need to prove the following inequality: for any$Z=(z_1,\dots,z_l)\in\mathbb{C}^l$for any$p\geq 2$and$l\geq 2$\begin{equation} \left|\left|\sum_{j=1}^l z_j\right|^p-\sum_{j=1}^l\left|z_j\right|^... 3answers 2k views ### Cauchy-Schwarz inequality for bilinear forms valued in an abstract vector space I previously posted this question on Math.SE but didn't receive an answer. It is perhaps a little vague; part of what I want to know is what question I should ask. First, consider the following form ... 1answer 2k views ### Can the fact that the square of an integer is a natural number be categorified? If$a$and$b$are natural numbers, then$a-b$is an integer and so the square$(a-b)^2$is a natural number. In particular $$(a-b)^2 \geq 0. \qquad (1)$$ Combining this fact with the identity$...
How to prove the following inequality: Let $X$ and $Y$ be $n\times m$ matrices with real entries. Prove that \begin{equation} \det\left(XY^T\right)^2 \leq \det\left(XX^T\right)\det\left(YY^T\right) . \...
I come across the following problem in my study. Let $x_i, y_i\in \mathbb{R}, i=1,2,\cdots,n$ with $\sum\limits_{i=1}^nx_i^2=\sum\limits_{i=1}^ny_i^2=1$, and $a_1\ge a_2\ge \cdots \ge a_n>0$. Is ...